Final answer:
The gauge pressure needed at the large end of a tapered pipe with a diameter twice as large at the small end can be found using Bernoulli's equation, which balances the kinetic, potential, and pressure energies of fluid flow.
Step-by-step explanation:
To find out what must be the gauge pressure at the large end to ensure water emerges at a speed of 12 m/s from the smaller end of a tapered pipe, which is elevated 8 m above the larger end, we can use the principle of conservation of energy, often applied as Bernoulli's equation in fluid dynamics. Given that the diameter of the large end is twice as large as the small end, we have a scenario where the cross-sectional area at the large end is four times that at the small end (since area is proportional to the square of the diameter).
To apply Bernoulli's equation, we would need to consider the kinetic energy (represented by the speed of the flow), the potential energy (represented by the height difference in gravitational field), and the pressure energy (the gauge pressure we want to find). Calculations would yield the pressure differential required to create such a flow speed difference given the height change.